German‚ 35 speak Italian‚ and 20 speak both German and Italian. How many embassy employees speak neither German nor Italian? Illustrate the situation with a Venn diagram. Answer: 3 11. Solve for [pic]: [pic] Answer: 8 Chapter 3: The Binomial Theorem 12. If [pic]‚ find n. Answer: 9 13. In the expansion of [pic]‚ find the following: a. The general term b. The term containing
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RISK THEORY - LECTURE NOTES 1. INTRODUCTION The primary subject of Risk Theory is the development and study of mathematical and statistical models to describe and predict the behaviour of insurance portfolios‚ which are simply financial instruments composed of a (possibly quite large) number of individual policies. For the purposes of this course‚ we will define a policy as a random (or stochastic) process generating a deterministic income in the form of periodic premiums‚ and incurring financial
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1 Gauss’ theorem Chapter 14 Gauss’ theorem We now present the third great theorem of integral vector calculus. It is interesting that Green’s theorem is again the basic starting point. In Chapter 13 we saw how Green’s theorem directly translates to the case of surfaces in R3 and produces Stokes’ theorem. Now we are going to see how a reinterpretation of Green’s theorem leads to Gauss’ theorem for R2 ‚ and then we shall learn from that how to use the proof of Green’s theorem to extend it
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and change is being found or created? Actually‚ this is a great philosophical question. Mathematics is like a religion. Though it appears to be a very precise and concrete subject with many proofs to support with‚ for example like the Pythagoras’s theorem to support the a relation in Euclidean geometry among the three sides of a right triangle‚ it is not always how it seems. We never know whether the so called proof is correct. What if one day somebody simply proved this theory wrong? There could be
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de physique‚ from her studies with Leibniz (Emilie). She then moved on to Newton. Algebraical Commentary was added to Newton’s’ book of principles of the mathematics. She also rewrote the book in French. She then studied under Pierre Louis de Maupertuis‚ a leading mathematicians and astronomers of the day. Maupertuis and Voltaire tried to get the French to move away from the Descartes and towards the ideas of Newton. The only problem with them moving to the new idea is that Newton’s work was
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MAPOA INSTITUTE OF TEGHNOLOGY Dep<rrtment of Mothemqtio vtst0N The l‚4apua lnstitute of Technology shall be a global center of excellence in education by providing instructions that are current in content and state-of-the-art in delivery; by engaging in cutting-edge‚ high impact research; and by aggressively taking on presen!day global concerns. Mrssr0N a. b. c. d. The [/apua Institute of Technology disseminates‚ generates‚ preserves and applies knowledge in various fields of study. The lnstitute
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An Important Note 28 29 33 36 39 41 44 46 48 50 52 55 56 3. Counting 3.1. 3.2. 3.3. 3.4. 3.5. Counting Lists Factorials Counting Subsets Pascal’s Triangle and the Binomial Theorem Inclusion-Exclusion 57 57 64 67 72 75 vi II How to Prove Conditional Statements 4. Direct Proof 4.1. 4.2. 4.3. 4.4. 4.5. Theorems Definitions Direct Proof Using Cases Treating Similar Cases 81 81 83 85 90 92 5. Contrapositive Proof 5.1. Contrapositive Proof 5.2. Congruence of Integers 5.3. Mathematical
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Homework sin5 xdx = − (1−u2)2du = − (u4−2u2+1)du cos5 x 2 cos3 x = + − cos x + C 5 3 − − − − − − − − − − − − − − − − − − − − −− u = cosn−1 x‚ v = cos x‚ v = sin x u = (n − 1) cosn−2 x(− sin x) cosn xdx = cosn−1 x sin x+(n−1) = cosn−1 x sin x+(n−1) cosn−2 x sin2 xdx cosn−2 x−(n−1) cosn xdx because sin2 x = 1 − cos2 x. Then 1 n−1 cosn xdx = cosn−1 x sin x+ cosn−2 xdx n n − − − − − − − − − − − − − − − − − − − − −− Apply the above identity for n = 3‚ we have cos2
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Rahul Chacko IB Mathematics HL Revision – Step One Chapter 1.1 – Arithmetic sequences and series; sum of finite arithmetic series; geometric sequences and series; sum of finite and infinite geometric series. Sigma notation. Arithmetic Sequences Definition: An arithmetic sequence is a sequence in which each term differs from the previous one by the same fixed number: {un} is arithmetic if and only if u n 1 u n d . Information Booklet u n u1 n 1d Proof/Derivation: u n 1
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UNIVERSITI TEKNOLOGI P E T RO NA S @t C OU R S E DATE TIME . GA B 2O13 B2133BUSINESS / SM STATISTTCS / QUANTITATIVE METHOD 27 MAY2008(TUESDAY) 2 .3 0PM- 5.30PM ( 3 hour s) INSTRUCTIONS CANDIDATES TO 1. 2. 3. 4. 5. Answer FIVE(5)outof SIX(6)questions theQuestion from Booklet. BeginEACHanswer a newpagein theAnswer on Booklet. lndicate clearly answers arecancelled‚any. that if Whereapplicable‚ showclearly stepstakenin arriving the solutions at and indicate ALL assumptions. Do notopenthisQuestion
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